4. Applications of Derivatives

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.

x¹⸍² and x¹⸍³

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.

ln x and log₁₀ x

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.

eˣ and 3ˣ

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.

2ˣ and 4ˣ⸍²

Cosine limits Let n be a positive integer. Evaluate the following limits.

lim_x→0 (1 - cos xⁿ) / x²ⁿ

Cosine limits Let n be a positive integer. Evaluate the following limits.

lim_x→0 (1 - cosⁿ x) / x²

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_u→ π/4 (tan u - cot u) / (u - π/4)

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_z→0 (tan 4z) / (tan 7z)

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 0 (1 - cos 3x) / 8x²

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 0 (sin² 3x) / x²

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→π (cos x +1 ) / (x - π )²

Two methods Evaluate the following limits in two different ways: Use the methods of Chapter 2 and use l’Hôpital’s Rule.

lim_x→0 (e²ˣ + 4eˣ - 5) / (e²ˣ - 1)

More limits Evaluate the following limits.

lim_x→1 (x ln x - x + 1) / (xln²x)

Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.

x² ln x; x³

Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.

x²⁰ ; 1.0001ˣ